Functional connections between bird eggshell stiffness and nest characteristics through risk of egg collision in nests

Abstract Eggs and nests are two critical traits for the ecological success of birds. Their functional interactions, however, remain unclear. Here, we examined the functional connections between egg stiffness and nest attachment, site and structure for 1350 avian species. We revealed high eggshell stiffness for eggs in nests with a pensile attachment, located on non‐tree vegetation or having a domed shape, suggesting that birds produce stiffer eggs in response to higher egg‐collision risk in unstable or enclosed nests. Interdependence models suggested that the evolution of eggshell stiffness was more likely to be driven by than drive that of nest characters. Our results implied a trade‐off between investment in competing for established nesting niches and producing stiff eggs to explore novel niches with high collision risk, possibly mediated by predation or thermoregulation. This study highlights an overlooked connection between nests and eggshells that may have broadened the ecological niches of birds.


Supplementary Results
Supplementary Tables   Table S1. Bird species and their nest and egg traits examined in the study. (in a separate Excel file) Table S2. Terminology and nomenclature in eggshell stiffness estimation. Table S3. Summary of a single PGLS model for examining the effects of the three nest characters on C number among the studied species.

Egg images processing
The images obtained from the Arctos database (http://arctos.database.museum) of the Museum of Vertebrate Zoology at UC Berkeley and The Book of Eggs (Hauber 2014) were initially 8-bit RGB in JPEG format, and were transformed into greyscale images with the pixel value estimated using the following formula: . After removing the salt and pepper noise with low-pass and median filters, the images were turned into binary images using Otsu's method (Otsu 1979) to contain only black and white. To make sure that the white area of the converted egg images by Otsu's method is the shape of an egg, we checked (i) the number of pixels of contiguous white to exclude the small white stains in the background of the image; (ii) a shape factor of the white area, defined as , where A stands for the number of pixels and p the perimeter of the white area¾the shape factor is 1 for circles and 0.78 for rectangles; and (iii) the eccentricity of the white area (distance between the two foci divided by the length of the long axis) ¾eccentricity is 0 for circles and 1 for rectangles. We then only retained images that have (i) the number of pixels of contiguous white > 300, (ii) a shape factor > 0.86, and (iii) the eccentricity > 0.1. To ensure that the sharp poles faced upwards, we estimated the second moment of inertia of an oval whose value was identical to the egg contour. That is, the long axis of the oval was the same as that of the egg, and hence the angle between the y-axis and the long axis of the oval was the angle to rotate for the egg profile. Then we divided the right half of the egg into five parts, producing eleven coordinates for the egg profile, and exported 0.299 0.587 0.114 them into a .txt file (Fig. S1). Images before and after image processing were exported for visual inspection to ensure that no mistakes were made during the processing.

Verifying the absolute eggshell stiffness estimated by the finite element method (FEM)
In engineering, the stiffness of a load-bearing structure can be determined or estimated by three distinct approaches: experimental, numerical, and theoretical. Among them, the numerical approach such as FEM is particularly powerful and has been extensively used in engineering. However, FEM is a method of approximation and its accuracy must be verified by experiments and theories. In a previous study (Juang et al. 2017), we first conducted a side-by-side comparison of the key physical and mechanical properties of several representative species, and found that characterization of eggshell stiffness using egg images and numerical simulations was consistent with that obtained by experiments using mechanical compression tests (See Fig. 1 and Table S1 in Juang et al. 2017). Second, we did a side-by-side comparison of eggshell stiffness of several representative species between FEM and shell theory, and found that the deviation was < 3%, which reaffirmed the accuracy of FEM (See Table S2 in Juang et al. 2017).

C number as a proper measure of eggshell stiffness for interspecific comparisons
The absolute stiffness of eggshell-K-is improper for interspecific comparison of egg resistance to external loads from other eggs or incubating birds due to the confounding effect of egg size. For example, the absolute stiffness of a massive elephant bird egg (Aepyornis maximus, K » 6.6´10 6 N m -1 ) is much larger than that of a hummingbird egg (Mellisuga minima, K » 1.1´10 4 N mm -1 ); however, it is unclear whether an elephant bird egg, compared to a hummingbird egg, is more likely to survive the collision of its neighboring eggs (induced by the incubating bird or other environmental forces), as an elephant bird egg is much heavier than a hummingbird egg and can cause more impact damage due to its higher momentum. To overcome this, Juang et al. (2017) developed a dimensionless metric, C number, to characterize eggs' stiffness with respect to egg mass and shape. Juang et al.
(2017) estimated the C numbers of the elephant bird and hummingbird eggs to be 11,993 and 17,400, respectively. That is, the elephant bird egg showed a similar or slightly lower level of crash resistance than the hummingbird egg without the confounding effects of egg weight or geometry-induced rigidity. Therefore, the C number facilitates the comparison of eggshell stiffness across the avian phylogeny and even over the evolutionary history.

Theoretical analysis and experimental verification of the effect of simulated compression direction on estimated C numbers
To compare the C number estimated from the absolute stiffness (K) simulated along the horizontal axis with that along the vertical axis, we conducted the compression simulation test with the egg sample positioned horizontally, with its equator in contact with the two rigid plates (see the normalized displacement. The subscript H denoted "horizontal" compression, compared to V denoting "vertical" compression. In this case, the indentation was locally identical to the indentation of a spherical shell of radius . Hence,  et al. 2017). The small difference was attributed to the fact that egg was not a perfect ellipsoid so some slight deviation in the simulation result was expected. In this study, we used the vertical Cv because it is relatively easy to construct the computer models without worrying about the problem of asymmetric convex shell, and CV and CH are good approximation.

Higher energy cost associated with higher C numbers
Although eggshell with high stiffness may prevent eggs from cracking, it is energetically costly. The stiffness of an ellipsoidal elastic shell is , where E, t, and n are the Young's modulus, the thickness, and the Poisson's ratio of the shell, respectively. k is the local curvature of the shell at the loading point (the pole of eggshell) and thus determined by the shape and size of eggshell. E and v are largely constant according to mechanical compression tests on 400 freshly laid intact eggs from 40 bird species, which belonged to 16 families and 11 orders (see Fig. 1h in Juang et al. 2017). Other things being equal, producing a stiffer egg (i.e., a larger K value) requires a thicker shell (i.e., a larger t value). Consequently, a higher C number ( ), which controls the effects of egg size and weight, also corresponds to a thicker shell, which led to a higher K value; a thicker shell requires more calcium, which is a critical but limited resource for reproduction in birds (Tilgar et al. 2002), among other materials, and thus is energy-expensive. Thus, natural selection may not always favor eggs with higher C numbers.

Definitions of nest character categories
We used three nest characters-site, structure, and attachment-to categorize the nests of the Ground sites referred to those nests built on the ground (including nests that were built over short grass and stably sat on the ground) or to the case where eggs were directly laid on the ground. Cliff/Bank sites indicated that the nests were built on cliffs, river banks, or rocks.
Underground sites indicated nests built in underground burrows. Water sites were the nests that were built by piling pebbles from the bottom of a water body to form an "island" and topping vegetation on the peak, or built on the elevated spots of sediment that emerge from the water, or set on (under)water plants and built above the water. For nest attachments, basally attached nests were those supported from their bottoms, including those stably located among interweaving branches or inside cavities. Laterally attached nests were those attaching their lateral part(s) to reeds, trees, or other objects except horizonal branches. Horizontally forked nests were supported by two or more horizontal tree or bush branches by their sides (not bottom). Pensile attachments indicated that the nests were hanged down from supporting objects and suspended in the air.

Evolutionary interdependence between nest characteristics and eggshell stiffness
Attachment ( Figure 3A): The correlated evolution model test showed strong support for a dependent model between nest attachment and C number over an independent model (Bayes Factor = 9.3). The attachment type of the ancestor of modern birds was more likely to be basal than non-basal.
On the other hand, the similar probabilities estimated for the ancestor with different levels of C number suggested that ancestral birds did not tend to have higher or lower stiffness than their offspring. The results also showed that for birds using basal attachment, their C number was more likely to transit from high values to low ones (q31 = 27.52) than vice versa (q13 = 20.78). For birds using non-basal attachment, their C number was more likely to transit from low values to high ones (q24 = 21.19) than vice versa (q42 = 16.33). The results were consistent with our finding that lower C numbers in extant birds using basal attachment than those using other attachment types (Fig. 2). Higher transition rates from non-basal attachment to basal attachment than vice versa in birds with either high or low C numbers might explain the prevalence of basal attachment in extant birds. The overall patterns regarding the influential roles of C number values and nest attachment types on each other imply that it is more likely that the evolution of C number depends on nest attachment types than vice versa.
Site ( Figure 3B): The correlated evolution model test showed strong support for a dependent model between nest site and C number over an independent model (Bayes Factor = 14). According to the dependent model, the low estimated probabilities of tree/non-tree vegetation nest sites for the root of the avian phylogeny suggested tree/non-tree-vegetation as derived traits relative to other nest sites. For birds using non-vegetation nest sites, their C number was more likely to transit from high to low values (q31 = 25.08) than vice versa (q13 = 10.19); for birds occupying tree/non-tree-vegetation nest sites, their C number was slightly more likely to change from low to high values (q24 = 27.59) than vice versa (q42 = 25.53). The results were consistent with our finding that higher C numbers in extant birds using tree or non-tree vegetation sites than those using other sites (Fig. 2). For birds with either high or low C numbers, they were more likely to transit from non-vegetation nest sites to tree/non-treevegetation sites than vice versa, although the transition rates were generally low (< 12). The differences in the transition rates may explain why more extant birds build nests on trees or non-tree vegetations than those on other nest sites. The overall patterns suggested that nest sites are more likely to determine the evolution of C number than the other way around.
Structure ( Figure 3C): The correlated evolution model test showed strong support for a dependent model between nest structure and C number over an independent model (Bayes Factor = 15.6). The high estimated probabilities of scrape/platform nests for the root of the avian phylogeny suggested this type of nest structure as an ancestral trait against other types. For birds using scrape/platform nests, their C numbers were more likely to transit from high to low values (q31 = 21.3) than vice versa (q13 = 9.27); for bird using nests with other structure types, the transition rate (q24 = 28.95) from low to high C numbers was slightly higher than that of the opposite direction (q42 = 26.96). The results were consistent with our finding that lower C numbers in extant birds using scrape/platform nests than those using other structure types (Fig. 2). For birds with either high or low C numbers, they were more likely to transit from scrape/platform nests to other nest structure types than vice versa, although the transition rates were generally low (< 7). This may explain why there are slightly fewer extant avian species building scrape/platform nests than those building cup, dome or cavity nests. The overall patterns supported that the evolution of C number was more likely to be driven by than drive that of nest structure. Tables   Table S2. Terminology and nomenclature in eggshell stiffness estimation. The definitions of the terms and how they were determined are provided. FEM stands for the finite element method, a computer simulation method routinely used by engineers to predict the stiffness of a structure, e.g., bridges, buildings, vehicles, and airplanes.

Terms Symbols Definition Determined by
The C number C Definition